Improve CM testing for elliptic curves over number fields

src/doc/en/reference/references/index.rst

@@ -1551,6 +1551,10 @@ REFERENCES:
             and Monographs*. American Mathematical Society,
             Providence, RI, 1999.
 
+.. [Cox1989] David A. Cox.
+         Primes of the form `x^2+ny^2`.
+         Wiley, 1989.
+
 .. [CK2008] Derek G. Corneil and Richard M. Krueger, *A Unified View
             of Graph Searching*, SIAM Jounal on Discrete Mathematics,
             22(4), 1259–-1276, 2008.
@@ -1647,6 +1651,11 @@ REFERENCES:
 
 .. [CrNa2020] \J.E. Cremona and F. Najman,  `\QQ`-curves over odd degree number fields, :arxiv:`2004.10054`.
 
+.. [CreSuth2023] \J.E. Cremona and A.V. Sutherland.
+                 *Computing the endomorphism ring of an elliptic curve
+                 over a number field*.
+                 :arxiv:`2301.11169`.
+
 .. [CoCo1] J.H. Conway, H.S.M. Coxeter
     *Triangulated polygons and frieze patterns*,
     The Mathematical Gazette (1973) 57 p.87-94
@@ -3699,6 +3708,11 @@ REFERENCES:
 .. [KMOY2007] \M. Kashiwara, K. C. Misra, M. Okado, D. Yamada.
               *Perfect crystals for* `U_q(D_4^{(3)})`, J. Algebra. **317** (2007).
 
+.. [Klaise2012] Janis Klaise.
+                *Orders in imaginary quadratic fields of small class number*
+                University of Warwick Undergraduate Masters thesis, unpublished (2012).
+                https://warwick.ac.uk/fac/cross_fac/complexity/people/students/dtc/students2013/klaise/janis_klaise_ug_report.pdf
+
 .. [KMR2012] \A. Kleshchev, A. Mathas, and A. Ram, *Universal Specht
              modules for cyclotomic Hecke algebras*,
              Proc. London Math. Soc. (2012) 105 (6): 1245-1289.
@@ -5860,6 +5874,12 @@ REFERENCES:
               Mathematics of Computation **80** (2011), pp. 477-500.
               :arxiv:`0809.3413v3`.
 
+.. [RouSuthZur2022] Jeremy Rouse, Andrew V. Sutherland, David Zureick-Brown.
+                    *`\ell`-adic images of Galois for elliptic curves over `\Q`* (and an appendix with John Voight).
+                    Forum of Mathematics, Sigma , Volume 10 , 2022.
+                    :doi: `10.1017/fms.2022.38`.
+                    :arxiv:`2106.11141`.
+
 .. [SV1970] \H. Schneider and M. Vidyasagar. Cross-positive matrices. SIAM
             Journal on Numerical Analysis, 7:508-519, 1970.
 
@@ -6177,6 +6197,11 @@ REFERENCES:
 .. [Watkins] Mark Watkins, *Hypergeometric motives over Q and their
              L-functions*, http://magma.maths.usyd.edu.au/~watkins/papers/known.pdf
 
+.. [Watkins2004] Mark Watkins.
+                 *Class numbers of imaginary quadratic fields*.
+                 Math. Comp. 73 (2004), 907-938.
+                 https://www.ams.org/journals/mcom/2004-73-246/S0025-5718-03-01517-5/
+
 .. [Wat2003] Joel Watson. *Strategy: an introduction to game
              theory*. WW Norton, 2002.
 

src/doc/en/thematic_tutorials/explicit_methods_in_number_theory/nf_galois_groups.rst

@@ -188,7 +188,7 @@ field).
 ::
 
     sage: w = [-1,-2,..,-200]
-    sage: v = [D for D in w if is_fundamental_discriminant(D)]
+    sage: v = [D for D in w if D.is_fundamental_discriminant()]
     sage: v
     [-3, -4, -7, -8, -11, -15, -19, -20, ..., -195, -199]
 
@@ -222,7 +222,7 @@ class number :math:`1`!
 ::
 
     sage: w = [1..1000]
-    sage: v = [D for D in w if is_fundamental_discriminant(D)]
+    sage: v = [D for D in w if D.is_fundamental_discriminant()]
     sage: len(v)
     302
     sage: len([D for D in v if QuadraticField(D,'a').class_number() == 1])

src/sage/quadratic_forms/binary_qf.py

@@ -607,7 +607,7 @@ def has_fundamental_discriminant(self):
             sage: Q.has_fundamental_discriminant()
             False
         """
-        return is_fundamental_discriminant(self.discriminant())
+        return self.discriminant().is_fundamental_discriminant()
 
     def is_primitive(self):
         r"""
@@ -1774,7 +1774,7 @@ def BinaryQF_reduced_representatives(D, primitive_only=False, proper=True):
 
     # For a fundamental discriminant all forms are primitive so we need not check:
     if primitive_only:
-        primitive_only = not is_fundamental_discriminant(D)
+        primitive_only = not D.is_fundamental_discriminant()
 
     form_list = []
 

src/sage/quadratic_forms/special_values.py

@@ -163,7 +163,7 @@ def QuadraticBernoulliNumber(k, d):
 
     Let us create a list of some odd negative fundamental discriminants::
 
-        sage: test_set = [d for d in range(-163, -3, 4) if is_fundamental_discriminant(d)]
+        sage: test_set = [d for d in srange(-163, -3, 4) if d.is_fundamental_discriminant()]
 
     In general, we have `B_{1, \chi_d} = -2 h/w` for odd negative fundamental
     discriminants::

src/sage/rings/integer.pyx

@@ -5590,7 +5590,8 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
         - ``proof`` (boolean, default ``True``) -- if ``False`` then
           for negative discriminants a faster algorithm is used by
           the PARI library which is known to give incorrect results
-          when the class group has many cyclic factors.
+          when the class group has many cyclic factors.  However the
+          results are correct for discriminants `D` with `|D|\le 2\cdot10^{10}`.
 
         OUTPUT:
 
@@ -5599,7 +5600,7 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
 
         .. NOTE::
 
-           This is not always equal to the number of classes of
+           For positive `D`, this is not always equal to the number of classes of
            primitive binary quadratic forms of discriminant `D`, which
            is equal to the narrow class number. The two notions are
            the same when `D<0`, or `D>0` and the fundamental unit of
@@ -6034,6 +6035,73 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
         """
         return self.__pari__().issquarefree()
 
+    def is_discriminant(self):
+        """
+        Returns True if this integer is a discriminant.
+
+        .. NOTE::
+
+            A discriminant is an integer congruent to 0 or 1 modulo 4.
+
+        EXAMPLES::
+
+            sage: (-1).is_discriminant()
+            False
+            sage: (-4).is_discriminant()
+            True
+            sage: 100.is_discriminant()
+            True
+            sage: 101.is_discriminant()
+            True
+
+        TESTS::
+
+            sage: 0.is_discriminant()
+            True
+            sage: 1.is_discriminant()
+            True
+            sage: len([D for D in srange(-100,100) if D.is_discriminant()])
+            100
+        """
+        return self%4 in [0,1]
+
+    def is_fundamental_discriminant(self):
+        """
+        Returns True if this integer is a fundamental_discriminant.
+
+        .. NOTE::
+
+            A fundamental discriminant is a discrimimant, not 0 or 1 and not a square multiple of a smaller discriminant.
+
+        EXAMPLES::
+
+            sage: (-4).is_fundamental_discriminant()
+            True
+            sage: (-12).is_fundamental_discriminant()
+            False
+            sage: 101.is_fundamental_discriminant()
+            True
+
+        TESTS::
+
+            sage: 0.is_fundamental_discriminant()
+            False
+            sage: 1.is_fundamental_discriminant()
+            False
+            sage: len([D for D in srange(-100,100) if D.is_fundamental_discriminant()])
+            61
+
+        """
+        if self in [0,1]:
+            return False
+        Dmod4 = self%4
+        if Dmod4 in [2,3]:
+            return False
+        if Dmod4 == 1:
+            return self.is_squarefree()
+        d = self//4
+        return d%4 in [2,3] and d.is_squarefree()
+
     cpdef __pari__(self):
         """
         Returns the PARI version of this integer.

src/sage/rings/number_field/number_field.py

@@ -12188,16 +12188,15 @@ def is_fundamental_discriminant(D):
     EXAMPLES::
 
         sage: [D for D in range(-15,15) if is_fundamental_discriminant(D)]
+        ...
+        DeprecationWarning: is_fundamental_discriminant(D) is deprecated; please use D.is_fundamental_discriminant()
+        ...
         [-15, -11, -8, -7, -4, -3, 5, 8, 12, 13]
         sage: [D for D in range(-15,15) if not is_square(D) and QuadraticField(D,'a').disc() == D]
         [-15, -11, -8, -7, -4, -3, 5, 8, 12, 13]
     """
-    d = D % 4
-    if d not in [0, 1]:
-        return False
-    return D != 1 and D != 0 and \
-        (arith.is_squarefree(D) or
-            (d == 0 and (D // 4) % 4 in [2, 3] and arith.is_squarefree(D // 4)))
+    deprecation(35147, "is_fundamental_discriminant(D) is deprecated; please use D.is_fundamental_discriminant()")
+    return Integer(D).is_fundamental_discriminant()
 
 
 ###################

src/sage/rings/number_field/number_field_element.pyx

@@ -4676,10 +4676,6 @@ cdef class NumberFieldElement_absolute(NumberFieldElement):
             sage: a.absolute_charpoly(algorithm='pari') == a.absolute_charpoly(algorithm='sage')
             True
         """
-        # this hack is necessary because quadratic fields override
-        # charpoly(), and they don't take the argument 'algorithm'
-        if algorithm is None:
-            return self.charpoly(var)
         return self.charpoly(var, algorithm)
 
     def absolute_minpoly(self, var='x', algorithm=None):
@@ -4707,10 +4703,6 @@ cdef class NumberFieldElement_absolute(NumberFieldElement):
             sage: b.absolute_minpoly(algorithm='pari') == b.absolute_minpoly(algorithm='sage')
             True
         """
-        # this hack is necessary because quadratic fields override
-        # minpoly(), and they don't take the argument 'algorithm'
-        if algorithm is None:
-            return self.minpoly(var)
         return self.minpoly(var, algorithm)
 
     def charpoly(self, var='x', algorithm=None):

src/sage/rings/number_field/number_field_element_quadratic.pyx

@@ -2129,10 +2129,16 @@ cdef class NumberFieldElement_quadratic(NumberFieldElement_absolute):
         # D = 1 mod 4
         return mpz_fdiv_ui(self.D.value, 4) == 1
 
-    def charpoly(self, var='x'):
+    def charpoly(self, var='x', algorithm=None):
         r"""
         The characteristic polynomial of this element over `\QQ`.
 
+        INPUT:
+
+        - ``var`` -- the minimal polynomial is defined over a polynomial ring
+           in a variable with this name. If not specified this defaults to ``x``
+        - ``algorithm`` -- for compatibility with general number field elements; ignored
+
         EXAMPLES::
 
             sage: K.<a> = NumberField(x^2-x+13)
@@ -2147,15 +2153,16 @@ cdef class NumberFieldElement_quadratic(NumberFieldElement_absolute):
         R = QQ[var]
         return R([self.norm(), -self.trace(), 1])
 
-    def minpoly(self, var='x'):
+    def minpoly(self, var='x', algorithm=None):
         r"""
         The minimal polynomial of this element over `\QQ`.
 
         INPUT:
 
-        -  ``var`` -- the minimal polynomial is defined over a polynomial ring
-           in a variable with this name. If not specified this defaults to
-           ``x``.
+        - ``var`` -- the minimal polynomial is defined over a polynomial ring
+           in a variable with this name. If not specified this defaults to ``x``
+        - ``algorithm`` -- for compatibility with general number field elements: and ignored
+
 
         EXAMPLES::
 
@@ -2651,11 +2658,18 @@ cdef class OrderElement_quadratic(NumberFieldElement_quadratic):
         """
         return ZZ(NumberFieldElement_quadratic.trace(self))
 
-    def charpoly(self, var='x'):
+    def charpoly(self, var='x', algorithm=None):
         r"""
         The characteristic polynomial of this element, which is over `\ZZ`
         because this element is an algebraic integer.
 
+        INPUT:
+
+        - ``var`` -- the minimal polynomial is defined over a polynomial ring
+           in a variable with this name. If not specified this defaults to ``x``
+        - ``algorithm`` -- for compatibility with general number field elements; ignored
+
+
         EXAMPLES::
 
             sage: K.<a> = NumberField(x^2 - 5)
@@ -2671,10 +2685,17 @@ cdef class OrderElement_quadratic(NumberFieldElement_quadratic):
         R = ZZ[var]
         return R([self.norm(), -self.trace(), 1])
 
-    def minpoly(self, var='x'):
+    def minpoly(self, var='x', algorithm=None):
         r"""
         The minimal polynomial of this element over `\ZZ`.
 
+        INPUT:
+
+        - ``var`` -- the minimal polynomial is defined over a polynomial ring
+           in a variable with this name. If not specified this defaults to ``x``
+        - ``algorithm`` -- for compatibility with general number field elements; ignored
+
+
         EXAMPLES::
 
             sage: K.<a> = NumberField(x^2 + 163)